Bilateral and Multilateral Exchanges for Peer-Assisted Content - - PowerPoint PPT Presentation

Bilateral and Multilateral Exchanges for Peer-Assisted Content Distribution

Christina Aperjis Social Computing Group HP Labs Joint work with Ramesh Johari (Stanford) and Michael J. Freedman (Princeton)

Christina Aperjis Bilateral vs. Multilateral Content Exchange 1

Peer-assisted content distribution

Users upload files to each other Work well only if users share files and upload capacity P2P systems try to incentivize users to share

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Bilateral and multilateral exchange

Most prevalent P2P exchange systems are bilateral: downloading is possible in return for uploading to the same user

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Bilateral and multilateral exchange

Most prevalent P2P exchange systems are bilateral: downloading is possible in return for uploading to the same user Drawback of Bilateral Exchange:

• nly works between users that have

reciprocally desired files

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Bilateral and multilateral exchange

Most prevalent P2P exchange systems are bilateral: downloading is possible in return for uploading to the same user Drawback of Bilateral Exchange:

• nly works between users that have

reciprocally desired files Multilateral exchange allows users to trade in more general ways but is more complex to implement (e.g., virtual currency)

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Bilateral and multilateral exchange

Most prevalent P2P exchange systems are bilateral: downloading is possible in return for uploading to the same user Drawback of Bilateral Exchange:

• nly works between users that have

reciprocally desired files Multilateral exchange allows users to trade in more general ways but is more complex to implement (e.g., virtual currency) Tradeoff: simplicity vs. participation

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Bilateral vs. multilateral

1 Comparison of equilibria

What are the efficiency properties

• f the allocations that arise at equilibria?

2 Quantitative comparison

What proportion of users cannot participate?

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Preliminaries

1 View content exchange as an economy:

Demand = download requests for content Supply = scarce system resources

2 What files do peers have?

We focus on exchange on a timescale

• ver which the set of files peers have remains constant.

3 Rates vs. bytes

We focus on download/upload rates, rather than total number of bytes transferred.

4 The network

In the model we study, the constraint is on upload capacity. More generally, a network structure may constrain uploads and downloads.

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Notation

i j rijf

rijf = upload rate of file f from i to j dif =

j rjif = download rate of f for peer i

ui =

j,f rijf = upload rate of peer i

vi(di, ui) = utility to peer i from (di, ui) Bi = bandwidth constraint of user i X = set of feasible rate vectors X = {r : r ≥ 0; ui ≤ Bi; rijf = 0 if i does not have file f }

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Bilateral content exchange

Peers exchange content on a pairwise basis Let Rij =

f rijf = rate of upload from i to j

Exchange ratio: γij = Rji/Rij As if there exist prices pij, pji, and all exchange is settlement-free: pijRij = pjiRji Thus: γij = pij/pji

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Bilateral content exchange

Peers exchange content on a pairwise basis Let Rij =

f rijf = rate of upload from i to j

Exchange ratio: γij = Rji/Rij As if there exist prices pij, pji, and all exchange is settlement-free: pijRij = pjiRji Thus: γij = pij/pji Peer i may be effectively price-discriminating (if pij = pik)

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Bilateral content exchange

Peers exchange content on a pairwise basis Let Rij =

f rijf = rate of upload from i to j

Exchange ratio: γij = Rji/Rij As if there exist prices pij, pji, and all exchange is settlement-free: pijRij = pjiRji Thus: γij = pij/pji Peer i may be effectively price-discriminating (if pij = pik) Example: BitTorrent Peer j splits upload rate Bj equally among kj peers with highest rates to j (the “active set”) For a peer i in the active set: γij =

Bj kjRij

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Multilateral content exchange

Users can trade a virtual currency, where downloading from peer j costs pj per unit rate Similar to an exchange economy

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Equilibria

In multilateral exchange, users optimize given prices Multilateral optimization max vi(di, ui) s.t.:

• j pjRji ≤ pi
• j Rij

r ∈ X

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Equilibria

In multilateral exchange, users optimize given prices Multilateral optimization max vi(di, ui) s.t.:

• j pjRji ≤ pi
• j Rij

r ∈ X In bilateral exchange, users

• ptimize given exchange ratios

Bilateral optimization max vi(di, ui) s.t.: Rji ≤ γijRij∀j r ∈ X

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Equilibria

In multilateral exchange, users optimize given prices Multilateral optimization max vi(di, ui) s.t.:

• j pjRji ≤ pi
• j Rij

r ∈ X In bilateral exchange, users

• ptimize given exchange ratios

Bilateral optimization max vi(di, ui) s.t.: Rji ≤ γijRij∀j r ∈ X At an equilibrium all users have optimized, and the market clears Multilateral equilibrium (ME) r∗ and prices p∗ Bilateral equilibrium (BE) r∗ and exchange ratios γ∗ Under mild conditions, both ME and BE exist

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Pareto efficiency

An allocation r is Pareto efficient if: no user’s utility can be strictly improved without strictly reducing another user’s utility ME are always Pareto efficient (First fundamental theorem of welfare economics) BE may not be Pareto efficient

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Pareto efficiency

When are BE efficient? Theorem Assume utility approaches −∞ as upload rate approaches capacity A BE (γ∗, r∗) is Pareto efficient if and only if there exists a supporting vector of prices p∗ such that (p∗, r∗) is a ME [Hard part to prove is the “only if”]

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Pareto efficiency: proof sketch

Given Pareto efficient BE (γ∗, r∗) find price vector p∗ such that (p∗, r∗) is a ME The proof exploits a connection between equilibria and reversible Markov chains Let R∗

ij = total rate from i to j at BE and R∗ ii = − j R∗ ij

For simplicity, suppose R∗ is an irreducible rate matrix of a continuous time MC (generalizes to nonirreducible case)

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Pareto efficiency: proof sketch

Let p be the unique invariant distribution of R∗ If R∗ is reversible, then: piR∗

ij = pjR∗ ji ⇒ γ∗ ij = pi/pj ⇒ BE ≡ ME

What is the intuition for this result? The invariant distribution gives a vector of prices at which agents could potentially trade When R∗ is reversible, agents’ trades balance on a pairwise basis with one vector of prices

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Pareto efficiency: proof sketch

What if R∗ is not reversible? pi

pj > γ∗ ij for some R∗ ij > 0

⇒ i “overpaid” to transact with j at BE

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Pareto efficiency: proof sketch

What if R∗ is not reversible? pi

pj > γ∗ ij for some R∗ ij > 0

⇒ i “overpaid” to transact with j at BE

• k γkiR∗

ki = k pk pi R∗ ki ⇒ i “underpaid” some j′

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Pareto efficiency: proof sketch

What if R∗ is not reversible? pi

pj > γ∗ ij for some R∗ ij > 0

⇒ i “overpaid” to transact with j at BE

• k γkiR∗

ki = k pk pi R∗ ki ⇒ i “underpaid” some j′

Can find cycle of users {1, ..., K} such that k “overpaid” k + 1 ∀k

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Pareto efficiency: proof sketch

What if R∗ is not reversible? pi

pj > γ∗ ij for some R∗ ij > 0

⇒ i “overpaid” to transact with j at BE

• k γkiR∗

ki = k pk pi R∗ ki ⇒ i “underpaid” some j′

Can find cycle of users {1, ..., K} such that k “overpaid” k + 1 ∀k Pareto improvement: Increase u∗

i and R∗ i,i−1 by ai

User i better off if ai+1

ai

> γ∗

i,i+1

Possible to find such ai’s, because

i γ∗ i,i+1 < 1

1 2 3 a a a

2 1 3

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Pareto efficiency: proof sketch

What if R∗ is not reversible? pi

pj > γ∗ ij for some R∗ ij > 0

⇒ i “overpaid” to transact with j at BE

• k γkiR∗

ki = k pk pi R∗ ki ⇒ i “underpaid” some j′

Can find cycle of users {1, ..., K} such that k “overpaid” k + 1 ∀k Pareto improvement: Increase u∗

i and R∗ i,i−1 by ai

User i better off if ai+1

ai

> γ∗

i,i+1

Possible to find such ai’s, because

i γ∗ i,i+1 < 1

1 2 3 a a a

2 1 3

r∗ Pareto efficient BE ⇒ R∗ reversible

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Pareto efficiency: proof sketch

Of course, in general R∗ may not be irreducible Instead, the graph of trades in the BE may have multiple connected components To complete the proof, we consider supporting price vectors p that arise as linear combinations of the unique invariant distributions on each component We show that if no supporting prices for the BE exist, then a cycle of agents (possibly spanning multiple connected components) can be found who have a Pareto improving trade

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Bilateral vs. multilateral

1 Pareto efficiency of equilibria 2 Participation

How many peers are able to trade bilaterally and multilaterally? We use a random model to quantify the density of trade produced by the two models

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Participation

Two peers are complementary if each has what the other wants A peer can trade bilaterally if she has a complementary peer A peer can trade multilaterally if it belongs on a cycle of peers along which peers want to trade

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Participation: asymptotic analysis

N users, K files Each user has one file to upload, and wants to download one file The probability a user wants or has the f -th most popular file is proportional to f −s (Zipf’s law)

s = 0: uniform popularity s > 1: popularity concentrated in relatively few files

Metric: expected proportion of users that cannot participate

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Participation: asymptotic analysis for s < 1

Let ρME (resp., ρBE) be the expected number of unmatched peers in multilateral (resp., bilateral) exchange Theorem When s ∈ [0, 1): If N > K 2, then ρBE → 0 If N < K 2, then ρBE ≥ (1 − s)2 If K log K < N, then ρME → 0 If N scales faster than K log K but slower than K 2, multilateral is significantly better than bilateral

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Participation: asymptotic analysis for s > 1

Theorem If s > 1, then ρBE → 0 for any scaling of K and N So in this case, bilateral performs very well Intuition: high concentration of popularity in a small number of files This result also holds:

when peers upload and download multiple files for more general random graph models

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BitTorrent popularity data

Dataset from [Piatek et al., 2008] 1.4M downloads, 680K peers, 7.3K files

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Data-driven comparison

What if we sample a random graph from the BT distribution?

2 4 6 8 10 x 10

5

0.2 0.4 0.6 0.8 1 number of users ρBE ρME

Multilateral exchange matches many more peers than bilateral

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Data-driven comparison

What if users can trade in triangles? Trilateral exchange converges much faster than bilateral

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Data-driven comparison

However, as the number of files a peer has increases, bilateral rapidly approaches multilateral

2 4 6 8 10 x 10

5

0.2 0.4 0.6 0.8 1 number of users ρBE m=1 m=2 m=5 m=10 m=20

m = # of files a peer has available for uploading

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Data-driven comparison

What if the number of files that users possess varies across different users?

2 4 6 8 10 x 10

5

0.4 0.5 0.6 0.7 0.8 0.9 1 number of users ρBE m=1 m ~ d m=2

d = distribution from dataset

• mean = 2.0084
• high variance

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Conclusions

A BE is Pareto efficient if and only if it corresponds to a ME Bilateral exchange performs very well in expectation if the file popularity is very concentrated and/or users share a sufficiently large number of files

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Conclusions

A BE is Pareto efficient if and only if it corresponds to a ME Bilateral exchange performs very well in expectation if the file popularity is very concentrated and/or users share a sufficiently large number of files Open issue: extend comparison to a dynamic setting, where downloads complete and preferences change over time users join and leave the system

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